Surrogate dimension reduction in measurement error regressions

We generalize the cumulative slicing estimator to dimension reduction where the predictors are subject to measurement errors. Unlike existing methodologies, our proposal involves neither nonparametric smoothing in estimation nor normality assumption on the predictors or measurement errors. We establish strong consistency and asymptotic normality of the resultant estimators, allowing that the predictor dimension diverges with the sample size. Comprehensive simulations have been carried out to evaluate the performance of our proposal and to compare it with existing methods. A dataset is analyzed to further illustrate the proposed methodology.